A Physics-Informed Neural Network with Hybrid Architecture for Magnetic Core Loss Prediction Under Complex Conditions

Abstract

Magnetic core loss is an important indicator for describing the performance of magnetic elements. The traditional physical model has an insufficient performance for predicting the magnetic core loss of magnetic elements under complex conditions such as high temperature, non-sinusoidal waveform, and high frequency. To address this issue, this study proposes a physics-informed neural network (PINN)-based model for magnetic core loss prediction. In particular, this PINN-based model is constructed with a hybrid network architecture as a baseline algorithm, which combines a convolutional long short-term memory network (Conv-LSTM), power spectral density (PSD), and an ensemble learning method (including extreme gradient boosting (XGB), gradient boosting regression (GBR), and random forest (RF)). This design aims to address the complexity of magnetic core loss prediction. Moreover, the Steinmetz equation (SE) is improved to enhance the adaptability under complex conditions, and this improved Steinmetz equation (ISE) is integrated as physical constraints embedded in the neural network for magnetic core loss prediction. Based on the traditional data-driven loss term, the physical residual term is introduced as a regularization constraint to enable the prediction to satisfy both the observed data distribution and physical law. The experimental results show that the PINN-based model has a good prediction performance of magnetic core loss under complex conditions.

1. Introduction

Magnetic core loss is an important indicator to describe the performance of magnetic elements. As high-frequency power electronics continue to evolve rapidly, magnetic core loss has become one of the key factors affecting the efficiency and operation cost of power electronics systems. The increase in magnetic core loss not only reduces energy efficiency but may also lead to overheating and significantly shorter service life [1,2,3]. Therefore, it becomes a core problem to realize high-precision prediction of magnetic core loss in magnetic elements during efforts to improve and refine modern power-electronics systems, and some scholars have researched this. Lotfi et al. [4] analyzed the challenges regarding the development of magnetic components for high-frequency applications, including magnetic core loss, parasitic effects, and heat dissipation management, and summarized the achievements of material improvement (e.g., MnZn and NiZn ferrite) and designed optimization (e.g., low profile and embedded structure). However, conventional approaches for estimating magnetic core loss, such as the Steinmetz-based model, are typically limited to sinusoidal excitation conditions because of their simplifying assumptions and ignore the complex effect of environmental factors, such as temperature, frequency, and waveform, on the loss, which limits its practical application [5]. For example, Barg et al. [6] reviewed the Steinmetz and Bertotti loss models, evaluated their advantages and disadvantages under symmetric magnetic flux density, and pointed out their insufficient accuracy under non-sinusoidal waveforms and dynamic conditions. In recent years, given the limitations of the Steinmetz equation (SE), researchers have proposed various improvement methods that significantly expand the model’s application scope. For the improved model under non-sinusoidal waveforms, Novak et al. [7] developed a prediction approach for magnetic losses in non-oriented electrical steel over wide temperature and frequency ranges, using an enhanced version of the SE, and reported a prediction deviation within 1.5%. Mu et al. [8] proposed the rectangular extension of the Steinmetz equation (RESE) model, which improved the prediction accuracy of the rectangular wave by introducing a parameter $\gamma$. Li et al. [9] proposed the generalized Steinmetz equation (GSE), aimed at better capturing the dependence of loss on the variation rate of magnetic flux density, and the prediction accuracy under non-sinusoidal waveforms was improved. Venkatachalam et al. [10] proposed an improved generalized Steinmetz equation (iGSE) to separate the primary and secondary hysteresis circuits and calculate the respective losses through a recursive algorithm. It achieves higher accuracy and wide applicability under non-sinusoidal waveforms. Yue et al. [11] compared a variety of magnetic core loss models and found that the waveform coefficient Steinmetz equation (WcSE) is better at low harmonics and 50% duty cycle, while iGSE has higher accuracy at extreme duty cycle and high frequency. These studies provide the foundation for the prediction of magnetic core loss in complex conditions. Ducharne and Sebald [12] explored the application of fractional derivatives in magnetic core loss modeling, and the result showed that the prediction accuracy under high frequency and complex waveforms is improved by introducing non-integer order dynamic terms. Barg and Bertilsson [13] proposed a method to model magnetic core losses corresponding to trapezoidal flux density waveforms, revealing the dynamic characteristic of relaxation loss, and developing a simplified model with an accuracy of over 96%. However, despite these improvements, these methods still have an insufficient prediction accuracy of magnetic core loss under complex conditions (such as special waveforms, high temperature, and high-frequency waveforms) [14,15]. At the same time, with the rapid development of data-driven technologies, the machine learning methods, which may capture nonlinear features and multivariate relationships of loss by the experimental data [16,17,18], are introduced into the modeling and prediction of magnetic core loss [1,19]. For example, Li et al. [20] proposed the MagNet framework, which accurately predicts magnetic core loss with a wavelet transform network and a convolutional neural network (CNN). Li et al. [21] have developed the MagNet open-source database, which includes 150,000 magnetic core loss data of various waveforms and ferrite materials, supports iGSE parameter extraction, neural network modeling, and other research. Li et al. [22] used the neural network to investigate how waveform characteristics, DC bias, and temperature jointly influence the magnetic core loss, and put forward the localized Steinmetz parameter calculation method, which significantly improves the loss prediction accuracy. Shen et al. [23] proposed a magnetic core loss estimation method based on a deep neural network (DNN), which realizes high-precision prediction with high-frequency error of less than 1% under high-frequency (up to 1 MHz) conditions by optimizing the network structure. Aslan et al. [24] used the adaptive neural fuzzy reasoning system (ANFIS) to efficiently predict the magnetic core loss of single-phase inverter transformers, with an average error of only 5.04%. On the other hand, data-driven machine learning models, while effective at capturing complex nonlinearities, often operate as “black boxes” without embedded physical laws. They typically require large volumes of reliable data and may suffer from limited generalization and suboptimal feature selection under complex operating conditions [25]. To address both the physical-model limitations and the black-box nature of data-driven methods, Raissi et al. [26] introduced physics-informed neural networks (PINNs), which embed governing equations as physical constraints in the loss function, enabling the network to learn simultaneously from data and physical laws. Fassi et al. [27] reviewed the application of the PINN model in the maintenance of power converters, emphasizing its advantages in improving the generalization ability and adaptability of models. Deng et al. [28] proposed a modeling method of magnetic core loss based on a knowledge-aware artificial neural network (KANN), which embeds the iGSE into the neural network, greatly improving the prediction accuracy and calculation speed. Solimene et al. [29] proposed a hybrid method combining physical understanding and machine learning, and MagNet data was used to predict magnetic core loss. Huang et al. [30] proposed a magnetization mechanism-inspired neural network (MMINN) based on the magnetization mechanism, which combines physical mechanisms and a data-driven method to achieve high precision. Recent studies have also demonstrated the effectiveness of PINN method for electromagnetic modeling and performance analysis of permanent-magnet devices, including PINN-based permanent magnet synchronous motor (PMSM) electromagnetism modeling, key-performance analysis of permanent-magnet couplers, and parametric modeling of PMSMs [31,32,33]. Although these studies proposed some PINN models, they mostly focus on predictions under conventional conditions, lacking extensive validation for different magnetic core materials and extreme conditions (such as non-sinusoidal waveforms, high frequency, and high temperature). This limits the adaptability of these methods under complex conditions.

Under this background, this study offers the following main contributions: (1) A PINN-based hybrid architecture for magnetic core loss prediction under complex operating conditions is developed. The architecture combines convolutional long short-term memory (Conv-LSTM) for temporal feature extraction, power spectral density (PSD) analysis for frequency-domain descriptors, and a tree-based ensemble learning module (Random Forest (RF), extreme gradient boosting (XGB), and Gradient Boosting Regression (GBR)), thereby enabling the effective fusion of time-domain, frequency-domain, and material-related features. (2) An improved Steinmetz-type core-loss model is formulated, in which temperature-, waveform-, and high-frequency-dependent correction terms are incorporated. The performance of this improved Steinmetz equation (ISE) is verified based on the MagNet dataset. (3) The ISE is embedded into the loss function as a physical constraint term. Based on the traditional data-driven loss term, the physical residual term is introduced as a regularization constraint to enable the prediction to satisfy both the observed data distribution and physical law.

The remainder of this paper is organized as follows: A PINN-based model for magnetic core loss prediction of magnetic elements is proposed in this study. Section 2 describes the definition of magnetic core loss of magnetic elements. Section 3 describes the proposed PINN-based approach for the magnetic core loss prediction. In particular, this model is constructed with a hybrid network architecture as the baseline algorithm. The SE is improved by introducing temperature, waveform, and high-frequency correction terms to expand the applicability under complex conditions, and this equation is integrated as physical constraints embedded in the neural network of the PINN-based model. A case study is conducted in Section 4 to assess the prediction performance of the magnetic core loss of the designed PINN-based model.

2. Magnetic Core Loss

Based on distinct physical origins, magnetic core losses within core materials are typically divided into three types: hysteresis-related, eddy-induced, and residual losses [34,35], as illustrated in Equation (1):

$$P_{core}=P_h+P_{cl}+P_e,$$

(1)

where $P_{core}$ represents the total core loss density (also known as unit volume core loss or simply core loss), and $P_h$, $P_{cl}$, and $P_e$ denote the hysteresis, eddy current, and residual components of the total core loss, respectively. Energy dissipation as heat occurs in the core during magnetization, resulting in hysteresis loss [36]. As depicted in Figure 1, in the magnetization curve, the shaded region illustrates the energy dissipated due to hysteresis effects occurring during magnetization. Key factors affecting hysteresis loss include peak magnetic flux density ($B_m$), remanence ($B_r$), maximum magnetic field strength ($H$), and coercivity ($H_c$). The formula for calculating hysteresis loss is given below:

$$P_h=k_h\cdot f\cdot B_m^{\beta },$$

(2)

where $f$ denotes the frequency, and $k_h$ and $\beta$ are parameters obtained by fitting to experimental measurements. Although magnetic core materials possess relatively high resistivity, they are still finite rather than ideal. Consequently, variations in magnetic flux induce internal voltages within the core, which in turn generate circulating eddy currents. These currents lead to energy loss in the form of heat dissipation within the material. The energy lost in this process is called eddy current loss, and its calculation is given by the following equation:

$$P_{cl}=k_{cl}\cdot f^2\cdot B_m^2,$$

(3)

where $k_{cl}$ is a data-driven coefficient linked to core geometry and material resistivity. Beyond hysteresis and eddy current losses, additional energy dissipation is categorized as residual loss. This type of loss is attributed to intrinsic material characteristics, including non-sinusoidal magnetic flux waveforms, localized flux variations, and complex inter-particle interactions [34]. The corresponding expression is given below:

$$P_e=8\cdot \sqrt{\rho \cdot S\cdot L\cdot V_0}\cdot f^{1.5}\cdot B_m^{1.5}$$

(4)

where $\rho$ is conductivity, $S$ is defined as the effective cross-section of the magnetic structure, $L$ is given as a constant (0.1356), and $V_0$ is fitted empirically.

3. PINN-Based Model for Magnetic Core Loss Prediction

3.1. Designed Network Architecture

To address the complexity in magnetic core loss prediction of magnetic elements, including challenges such as high-frequency, non-sinusoidal waveform scenarios, temperature variations, and material property dependencies, this study proposes a PINN-based model for the magnetic core loss prediction. In particular, this PINN-based model is constructed with a hybrid network architecture as the baseline algorithm, and the ISE model is integrated, serving as physical constraints integrated into the neural architecture used for estimating magnetic core losses. The hybrid architecture is presented in Figure 2, which combines a Conv-LSTM network, PSD, and an ensemble learning method [37,38] (such as XGB, GBR, and RF). In supervised learning, an ensemble learning framework refers to a structured strategy that integrates multiple base learners to obtain a final prediction that is more accurate and robust than any individual model.

To fully represent the influencing factors of core loss, signal features were derived across time and frequency dimensions. Temporal indicators [39,40] like standard deviation, mean, skewness, and kurtosis describe how magnetic flux density evolves. Frequency-domain features [41,42], including spectral entropy and harmonic frequencies, reveal the signal energy distribution and high-frequency contributions to magnetic core loss. By leveraging a Conv-LSTM network [43,44] for time-domain feature extraction and utilizing PSD [45,46] for frequency-domain analysis, the model effectively captures multidimensional signal characteristics, providing a comprehensive representation of loss attributes. To reduce redundancy in high-dimensional features and improve computational efficiency, the principal component analysis (PCA) method [47] was employed for dimensionality reduction. A unified feature representation was obtained by merging the reduced descriptors extracted from both temporal and spectral domains. This feature fusion strategy retains critical information and mitigates the risk of overfitting. Moreover, during the model design phase, three regression algorithms were selected: RF [48], XGB [49], and GBR [50]. These algorithms were chosen for their respective advantages. For example, the RF offers strong resistance to overfitting by constructing multiple decision trees; the XGB excels in capturing complex nonlinear relationships through gradient boosting; and the GBR demonstrates superior performance when handling small sample data. For the ensemble learning framework, cross-validation was used to evaluate the base learners and to tune their hyperparameters (e.g., number of trees, learning rate, etc.) for good ${\theta }_{\mathrm{RF}}^{*}, {\theta }_{\mathrm{XGB}}^{*}, {\theta }_{\mathrm{GBR}}^{*}$ performance on the training data. A stacking ensemble, rather than majority voting or simple averaging, was adopted to combine the outputs of the base learners. In the first level, the three base learners produce predictions for a sample with a feature vector $x_i$:

$${\widehat{y}}_{i,\mathrm{RF}}^{(1)}=f_{\mathrm{RF}}(x_i;{\theta }_{\mathrm{RF}}^{*}),\hspace{1em}{\widehat{y}}_{i,\mathrm{XGB}}^{(1)}=f_{\mathrm{XGB}}(x_i;{\theta }_{\mathrm{XGB}}^{*}),\hspace{1em}{\widehat{y}}_{i,\mathrm{GBR}}^{(1)}=f_{\mathrm{GBR}}(x_i;{\theta }_{\mathrm{GBR}}^{*}).$$

(5)

These outputs are concatenated to form the meta-feature vector below:

$$z_i={\left[\begin{array}{ccc}{\widehat{y}}_{i,\mathrm{RF}}^{(1)},& {\widehat{y}}_{i,\mathrm{XGB}}^{(1)},& {\widehat{y}}_{i,\mathrm{GBR}}^{(1)}\end{array}\right]}^{\top }.$$

(6)

In the second level, a linear regression model is used as the meta-learner to map $z_i$ to the final ensemble prediction:

$${\widehat{y}}_i=w_0+w_1{\widehat{y}}_{i,\mathrm{RF}}^{(1)}+w_2{\widehat{y}}_{i,\mathrm{XGB}}^{(1)}+w_3{\widehat{y}}_{i,\mathrm{GBR}}^{(1)},$$

(7)

where $w_0,w_1,w_2,w_3$ are the parameters learned from data. To avoid leakage in the stacking procedure, the meta-learner was trained using out-of-fold predictions generated from five-fold splits of the training set. Base learners were fitted on four folds and used to predict the held-out fold, and the aggregated out-of-fold predictions were used to train the meta-learner. For final evaluation, base learners were refitted using the full training set, and their test-set predictions were subsequently provided to the fixed meta-learner to obtain the final leakage-free ensemble output. The hyperparameters of the machine-learning and ensemble components were determined via grid-search optimization on the training subset, with five-fold cross-validation used for robust performance estimation. The configuration yielding the lowest validation error was selected as the final setting. Moreover, to evaluate how well the PINN-based model forecasts the output, standard error metrics, including root mean square error (RMSE), mean squared error (MSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the coefficient of determination (R²), were applied.

3.2. Embedding of Physical Information

PINNs are generally characterized by a training objective that combines a data-fitting loss with a physics-residual term. This residual is generally constructed from a governing partial or ordinary differential equation (PDE/ODE). In the classical PDE-based formulation, a PINN typically embeds the governing equation in the following form:

$$\mathcal{N}(u(x))=0,\hspace{1em}x\in \mathrm{\Omega },$$

(8)

where $\mathcal{N}(\cdot )$ denotes a differential operator (PDE/ODE) derived from physics. The network $u_{\theta }(x)$ is trained by minimizing a composite loss consisting of a data-misfit term and a physics-residual term:

$$\mathcal{L}(\theta )={\lambda }_d{\mathcal{L}}_{data}(\theta )+{\lambda }_p{\mathcal{L}}_{phy}(\theta ),\hspace{1em}{\mathcal{L}}_{phy}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\Vert \mathcal{N}(u_{\theta }(x_i))\Vert }^2.$$

(9)

In addition, in the broader physics-informed/physics-guided machine-learning framework, the same residual-minimization principle can also be instantiated using constitutive laws or other empirically validated algebraic relations, rather than explicit PDEs [51,52]. This study follows this broader notion of PINNs. Instead of enforcing a Maxwell-type differential equation for the electromagnetic field, the physical constraints of the ISE model are incorporated into a PINN framework to estimate magnetic core loss. The ISE model is constructed to consider temperature and waveform distortion, while its coefficients are calibrated on the training subset of the MagNet dataset and then kept fixed during network training. In this way, the proposed PINN-based model can enable the prediction to satisfy both the observed data distribution and physical law. Specifically, the SE is a classical calculation model of magnetic core loss of a magnetic element used to describe the relationship between magnetic core loss and working frequency and magnetic flux density. However, this equation has some limitations in practical applications, especially in non-sinusoidal waveforms, high frequency, and complex conditions, and the prediction results often deviate from the actual loss. For this purpose, the SE is improved to accommodate complex conditions and improve the prediction accuracy. The form of the traditional SE can be expressed as follows:

$$P_{core}=k\cdot f^{\alpha }\cdot B_m^{\beta },$$

(10)

where the parameters $k$, $\alpha$, and $\beta$ are determined based on experiments. This formula assumes a sinusoidal waveform, neglecting temperature, material properties, and the influence of non-sinusoidal excitation on core loss, making it unsuitable for complex operating conditions.

The classical model, based solely on the assumption of sinusoidal excitation, neglects several critical physical factors. Specifically, the variations in $B_{sat}$ and $H_c$ caused by temperature are typically overlooked in magnetic material modeling, as well as additional losses caused by harmonics under non-sinusoidal waveforms and the interaction caused by skin effect and proximity effect under high-frequency conditions. In practical applications, magnetic cores must endure diverse and complex operational environments. These include large temperature variations, complex non-sinusoidal excitation waveforms, and broad frequency ranges. Under such circumstances, the interactions among temperature, waveform shape, and frequency become highly nonlinear. As a result, the prediction error of the single-power-law model increases significantly. To address this issue, three physically meaningful correction terms have been introduced into the classical loss equation: a temperature-dependent power-law term $T^{\tau }$, a waveform factor $H$ that quantifies the harmonic-to-fundamental power ratio, and a high-frequency correction term ${\widehat{f}}_{high}$. These modifications are aimed at improving prediction accuracy under diverse operating conditions. The rationality of these assumptions is later validated through experimental results in this study, which confirm that core loss exhibits a log-linear relationship with temperature. Specifically, in a log-log plot, the natural logarithm of core loss $\ln P_{core}$ versus the natural logarithm of temperature $\ln T$ forms a straight line with a stable slope, supporting the inclusion of the power-law term $T^{\tau }$ for capturing thermal effects.

Therefore, the ISE is proposed based on the consideration of temperature, excitation waveform, and frequency correlation. Specifically, the temperature correction factor is introduced to account for the influence of temperature on the saturated flux density and coercive force, aiming to improve the accuracy of loss correction and better characterize the loss behavior of the magnetic core. Meanwhile, since magnetic core loss is already known to follow power-law dependencies on excitation frequency $f$ and peak flux density $B_{\max }$, temperature is likewise incorporated into the same empirical framework. In this study, the temperature correction factor $T$ is introduced in the loss calculation formula, and the corrected magnetic core loss expression is as follows:

$$P_{core}=k\cdot f^{\alpha }\cdot B_{\max }^{\beta }\cdot T^{\tau },$$

(11)

where $\tau$ denotes the material-dependent coefficient for temperature adjustment. In practical applications, non-sinusoidal waveform excitation conditions can significantly increase magnetic core loss. To quantify the effect of harmonic components on the loss, the harmonic content factor $H$ is introduced. The corrected magnetic core loss expression is as follows:

$$P_{core}=k\cdot f^{\alpha }\cdot B_{\max }^{\beta }\cdot (1+H),$$

(12)

where $H$ represents the proportion of the harmonic energy in the total energy. The loss properties of magnetic materials exhibit nonlinear changes with increasing frequency. Therefore, for the high-frequency conditions, the frequency nonlinear correction term $\delta$ and the high-frequency weight ${\widehat{f}}_{high}$ are introduced to capture the impact of high frequency on magnetic core loss, with the magnetic core loss equation described as follows:

$$P_{core}=k\cdot f^{\alpha }\cdot (1+\delta {\widehat{f}}_{high})\cdot B_{\max }^{\beta }$$

(13)

where $\delta$ is the high-frequency correction coefficient, and ${\widehat{f}}_{high}=f/f_0$ is the weight of the high-frequency part, where $f_0$ represents the lowest frequency used as the reference frequency. Based on the above corrections, the ISE proposed in this study is as follows:

$$P_{core}=k\cdot T^{\tau }\cdot f^{\alpha }\cdot B_{\max }^{\beta }\cdot (1+H)\cdot (1+\delta {\widehat{f}}_{high}).$$

(14)

The ISE model can be regarded as a physics-inspired law, its functional form reflects trends of core loss with respect to frequency, peak flux density, temperature, and harmonic content. Furthermore, the loss function for the physical residual term is formulated by combining the prediction errors with a physical constraint derived from the ISE. The loss function calculates the errors between the predicted core losses and the actual core losses. The corresponding loss function is shown in Equation (15):

$${\mathcal{L}}_{ISE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(P_{pred}^{(i)}-P_{core}^{(i)}\right)}^2},$$

(15)

where $P_{core}^{(i)}$ is the $i$th loss value calculated by the ISE. To enhance robustness to noise and improve generalization under imperfect data, an additional MAE term is introduced alongside the baseline MSE term. Specifically, the MSE term is used to measure the squared deviation from the observed values, while the MAE term focuses on the absolute deviation, as shown in Equations (16) and (17). This combination helps improve the ability of the model to generalize, especially under noisy or imperfect data conditions:

$${\mathcal{L}}_{\mathrm{MSE}}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{(P_{pred}^{(i)}-P_{actual}^{(i)})}^2,$$

(16)

where $P_{pred}^{(i)}$ is the $i$th magnetic core loss predicted by the model; $P_{actual}^{(i)}$ is $i$th actual loss of experimental observation. N indicates the sample size, and ${\mathcal{L}}_{MAE}$ quantifies the mean absolute error to measure the deviation from the actual value. The equation is as follows:

$${\mathcal{L}}_{\mathrm{MAE}}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}|P_{pred}^{(i)}-P_{actual}^{(i)}|.$$

(17)

The final loss function integrates the MSE, MAE, and ISE terms by a weighted summation, as shown in Equation (18). This comprehensive formulation ensures that the model simultaneously achieves high data-fitting accuracy and maintains consistency with physical principles:

$$\mathcal{L}={\lambda }_1\cdot {\mathcal{L}}_{MSE}+{\lambda }_2\cdot {\mathcal{L}}_{MAE}+{\lambda }_3\cdot {\mathcal{L}}_{ISE},$$

(18)

where ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are the weights of each term in the loss function, respectively. The physics-informed training objective was formulated as a weighted combination of data-fitting losses and a physics constraint term. Specifically, the final objective integrates the MSE term, the MAE term, and the physics constraint term by a weighted summation (Equation (18)). The weights ${\lambda }_1,{\lambda }_2,{\lambda }_3$ were treated as tunable hyperparameters rather than being manually assigned. A grid-search strategy was adopted, where each weight was searched within [0, 1] using a step size of 0.1. For each candidate weight setting, model parameters were fitted on the training set, and the optimal weight configuration was selected by minimizing the loss value of the same total objective defined in Equation (18).

The magnetic core loss prediction procedure based on the PINN-based model is shown in Figure 3. First, features extracted from both frequency and time domains, along with material-related attributes, are extracted to provide a comprehensive representation of the magnetic behavior under various operating conditions. The designed network architecture is composed of Conv-LSTM-based temporal modeling, feature fusion mechanisms, and ensemble learning methods (including RF, XGB, and GBR), allowing for complex nonlinear relationships to be effectively captured. An improved version of the SE is introduced, incorporating correction terms for temperature, waveform distortion, and high-frequency effects, thereby providing a physics-informed prior that enhances prediction accuracy. Subsequently, a PINN-based model is constructed for magnetic core loss prediction, in which the loss function integrates empirical prediction errors (MSE, MAE) with a physical constraint term (ISE model). To validate the performance of the proposed approach, a case-based analysis is implemented.

4. Case Study Validation

4.1. Experimental Data

Magnetic core loss data were obtained using the alternating current (AC) power method [53,54]. Excitation and induction windings were uniformly wound on a toroidal core, a periodic signal generated by a function generator was amplified by a high-frequency power amplifier and applied to the excitation winding, and steady-state terminal voltage $u\left(t\right)$ and current $i\left(t\right)$ were recorded to compute the unit-volume loss $P_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ with geometric parameters. By energy conservation, $P_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ admits equivalent expressions as a time-integrated power density and as the hysteresis-loop area:

$$P_{core}={\displaystyle \frac{1}{T_p}}{\displaystyle {\int }_0^{T_p}u}(t)\cdot i(t)dt/\left(A_e\cdot l_e\right)={\displaystyle \frac{1}{T_p}}{\displaystyle {\int }_{B(0)}^{B(T)}H}dB,$$

(19)

where $T$ is the period, $A_e$ and $l_e$ denote the effective core area and the average magnetic path length, and $H$ and $B$ are the magnetic field strength and flux density, respectively. The dataset was sourced from the MagNet open-source database on the online platform available at https://mag-net.princeton.edu/ (accessed on 5 October 2024) using an automated measurement platform with precise waveform control, temperature regulation, and high-resolution acquisition [21,55,56,57]. Four widely used ferrites were considered: TDK N87, TDK N27, Fair-Rite 77, and Ferroxcube 3C94, with sample counts of 3400, 3000, 3200, and 2800, respectively, totaling approximately 12,400 records. Operating conditions covered temperatures of 25, 50, 70, and 90 °C, a frequency range of 50–500 kHz, sinusoidal/triangular/trapezoidal excitation waveforms, and a peak flux density $B_m$ constrained to 0.01–0.30 T. To preserve resolution in rapidly varying regions while avoiding oversampling in flat regions, this study adopted 10–20 logarithmically spaced samples per frequency decade. To reduce factor coupling and highlight the main effects of material and waveform variables, the duty cycle was fixed. Material properties, device geometry (e.g., R-type and toroidal), and relative permeability are summarized in

Table 1. Summary of magnetic core materials.

Material	Manufacturer	Application	μr	Tested Core
N87	TDK Electronics (Munich, Germany)	Power transformers	2200	R34.0X20.5X12.5
N27	TDK Electronics (Munich, Germany)	Power transformers	2000	R20.0X10.0X7.0
77	Fair-Rite (Wallkill, NY, USA)	High/low flux inductive designs	2000	5977001401
3C94	Ferroxcube (New Taipei City, Taiwan)	Power and general-purpose transformers	2300	TX-20-10-7

to ensure consistency between modeling assumptions and experimental setup, and the measurement workflow and data preprocessing are illustrated in Figure 4. Figure 5 summarizes, via box-and-whisker plots, the amplitude distributions of excitation waveforms (sinusoidal, triangular, trapezoidal) for the tested materials across 25–90 °C and multiple frequencies. The distributions are compact and balanced with no apparent outliers, indicating cross-temperature and cross-frequency consistency suitable for downstream analyses. Figure 6 presents representative time-domain magnetic-flux-density traces for the four materials, selecting for each waveform the lowest and highest frequency at each temperature; the traces retain the expected sinusoidal, triangular, and trapezoidal profiles, evidencing stable waveform fidelity across operating conditions and enabling like-for-like comparisons.

4.2. Feature Analysis

Magnetic core loss of magnetic element is a complex physical phenomenon that is correlatively affected by many factors, such as time domain characteristics, frequency domain characteristics, and material properties. The original high-dimensional and unstructured data are transformed into key features in this section that reflect the physical nature from the time and frequency domains through feature extraction, thus improving the model’s ability to describe the data. Seven features are extracted from the time series of 1024 sampling points in one cycle, including mean, standard deviation, skewness, kurtosis, peak, peak-to-peak, and the sum of squares of magnetic flux density. The mean value of magnetic flux density, which reflects its overall magnitude, can be derived using the formula below:

$${\overline{B}}_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{B}_{ij}},$$

(20)

where $N$ denotes the total number of sampling points, $B_{ij}$ indicates the magnetic flux density at the $j$th sampling position of the $i$th sample, and ${\overline{B}}_i$ stands for the mean magnetic flux density of the $i$th sample. The standard deviation, defined as the square root of variance, quantifies the dispersion of flux density values across different sampling locations. The corresponding formula is given below:

$${\sigma }_i=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{\left(B_{ij}-{\overline{B}}_i\right)}^2}},$$

(21)

where ${\sigma }_i$ denotes the standard deviation of the flux density corresponding to the $i$th sample. Skewness quantifies both the direction and extent of deviation in the magnetic flux density profile and characterizes the asymmetry inherent in its distribution. The relevant formula is provided as follows:

$$S{K}_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{\left(\frac{B_{ij}-{\overline{B}}_i}{{\sigma }_i}\right)}^3},$$

(22)

where $S{K}_i$ denotes the skewness associated with the flux density in the $i$th sample. Kurtosis, also referred to as the peak coefficient, is applied to examine the normality of the flux density distribution and to quantify the sharpness of its peak. The corresponding formula is given below:

$$K_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{\left(\frac{B_{ij}-{\overline{B}}_i}{{\sigma }_i}\right)}^4},$$

(23)

where $K_i$ represents the kurtosis of the magnetic flux density in the $i$th sample. The peak value is the maximum value of the absolute magnetic flux density among the 1024 sampling points, and the calculation formula is as follows:

$$B_{mi}=\max \left(\left|B_{ij}\right|\right),$$

(24)

where $B_{mi}$ denotes the maximum magnetic flux density observed in the $i$th sample. The peak-to-peak magnitude refers to the variation between the highest and lowest values of magnetic flux density within one cycle, reflecting the extent of fluctuation in flux density under the given condition. The corresponding calculation is as follows:

$$B_{ipp}=B_{imax}-B_{imin},$$

(25)

where $B_{imax}$ represents the peak of magnetic flux density in $i$th sample, $B_{imin}$ represents the minimum magnetic flux density value for $i$th sample, and $B_{ipp}$ represents the peak value of magnetic flux density in $i$th sample. Use $E_i$ to represent the sum of the square of magnetic flux density over a period with the following formula:

$$E_i={\displaystyle \sum _{j=1}^N{B}_{ij}^2},$$

(26)

where $E_i$ represents the sum of square magnetic flux density for the $i$th sample.

Secondly, the spectral data is obtained by Fourier transform. When the magnetic flux density $B_{ij}$ of the $j$th sampling point of the $i$th sample is $X(B_{ij})$ after Fourier change, four variables are extracted, including harmonic frequency, spectral entropy, the standard deviation of channel bandwidth, and the spectral energy. The harmonic order, which is the ratio of the harmonic frequency to the base wave frequency, is an integer. The calculation formula is as follows:

$$c_i={\displaystyle \frac{f_i}{{\widehat{f}}_i}},$$

(27)

where $f_i$ is the harmonic frequency of the $i$th sample, ${\widehat{f}}_i$ is the fundamental frequency of the $i$th sample, and $c_i$ is the harmonic order of the $i$th sample. Spectral entropy is mainly used to describe the spectral distribution characteristics of magnetic flux density. The calculation formula is as follows:

$$H_i=-{\displaystyle \sum _{j=1}^N{n}_j}{\log }_2{n}_j,$$

(28)

where $n_j={\displaystyle \frac{{\left|X\left(B_{ij}\right)\right|}^2}{{\displaystyle \sum _{j=1}^N{\left|X\left(B_{ij}\right)\right|}^2}}}$ represents the probability distribution of each frequency component, and $H_i$ represents the spectral entropy of the $i$th sample. Channel bandwidth refers to the width of the signal frequency spectrum, that is, the wavelength or the range of the signal frequency, and its calculation formula is as follows:

$$W_i=\sqrt{f_{i2}-f_{i1}},$$

(29)

where $f_{i2}$ represents the highest frequency the $i$th sample channel can pass, $f_{i1}$ is the lowest frequency the $i$th sample channel can pass, and $W_i$ is the standard deviation of the channel bandwidth of the $i$th sample. The spectral energy is the sum of squares of the data after the Fourier transformation, and the calculation formula is as follows:

$$E{X}_i={\displaystyle \sum _{j=1}^N{\left|X\left(B_{ij}\right)\right|}^2},$$

(30)

where $E{X}_i$ represents the spectral energy of the $i$th sample. According to the above equation, the results of the feature extraction of the experimental data used in this study are shown in

Table 2. Results of feature extraction analysis (Part 1: basic features).

ID	Mean Value	Standard Deviation	Skewness	Kurtosis	Peak Value	Peak-to-Peak Value
1	−4.004 × 10−11	2.040 × 10−2	−6.627 × 10−4	1.502	2.885 × 10−2	5.769 × 10−2
2	−2.539 × 10−11	2.223 × 10−2	−1.222 × 10−3	1.502	3.143 × 10−2	6.285 × 10−2
3	−9.765 × 10−13	2.511 × 10−2	−4.942 × 10−4	1.503	3.554 × 10−2	7.105 × 10−2
.	.	.	.	.	.	.
.	.	.	.	.	.	.
.	.	.	.	.	.	.
12398	−7.813 × 10−11	2.973 × 10−2	1.486 × 10−2	1.671	4.888 × 10−2	9.681 × 10−2
12399	6.348 × 10−11	3.338 × 10−2	1.300 × 10−2	1.673	5.489 × 10−2	1.088 × 10−1
12400	−4.688 × 10−11	4.203 × 10−2	1.475 × 10−2	1.671	6.914 × 10−2	1.369 × 10−1

and

Table 3. Results of feature extraction analysis (Part 2: remaining features).

ID	Energy	Harmonic Order	Entropy Spectral	Channel Bandwidth	Spectral Energy
1	0.427	1	5.262 × 10−5	3.214	2.182 × 102
2	0.506	1	6.007 × 10−5	3.076	2.589 × 102
3	0.645	1	7.310 × 10−5	2.859	3.305 × 102
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
12398	0.905	7	4.117 × 10−1	7.7389	4.633 × 102
12399	1.141	7	4.106 × 10−1	7.633	5.840 × 102
12400	1.809	7	4.107 × 10−1	7.450	9.263 × 102

.

4.3. Verification of the Improved Steinmetz Equation

To validate the reliability of the ISE embedded in the PINN-based model, multiple kinds of magnetic core materials were tested under multiple waveforms (including sinusoidal, triangular, and trapezoidal) and a randomly selected working frequency. The relationship between various temperatures and magnetic core loss was investigated, with temperature taken as the abscissa and magnetic core loss as the ordinate, as shown in Figure 7. For each material, five working frequencies were randomly chosen under identical conditions. The magnetic core loss $P_{core}$ varies with temperature and shows a continuous decrease in a power-law manner as temperature increases. For both triangular (Figure 7e) and trapezoidal excitations (Figure 7f), the predicted loss decreases monotonically with increasing temperature and exhibits a frequency ordering consistent with that observed in the sinusoidal case.

To quantitatively assess the performance of different Steinmetz-type loss formulations independently of the PINN architecture, four kinds of magnetic core materials were tested under multiple waveforms and randomly selected operating frequencies, namely, the classical SE, the Modified Steinmetz Equation (ModSE) [58], the iGSE [10], and the proposed ISE model. For non-sinusoidal excitation, the ModSE introduces an equivalent frequency $f_{\mathrm{eq}}$ to extend the classical SE. The volumetric core loss is written as follows:

$$P_{core}=\left(\begin{array}{c}k{f}_{\mathrm{eq}}^{\alpha -1}{B}_m^{\beta }\end{array}\right)f_r,$$

(31)

where $f_r$ is the fundamental (base) frequency of the non-sinusoidal waveform. The equivalent frequency is defined using the following equation:

$$f_{\mathrm{eq}}={\displaystyle \frac{2}{\mathrm{\Delta }{B}^2{\pi }^2}}{\displaystyle {\int }_0^T{\left(\frac{\mathrm{d}B(t)}{\mathrm{d}t}\right)}^2}\mathrm{d}t,\hspace{1em}\mathrm{\Delta }B=B_{\max }-B_{\min }.$$

(32)

The iGSE expresses the loss directly as a time integral of $\mathrm{d}B(t)/\mathrm{d}t$:

$$P_{core}={\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T{k}_i}{\left|{\displaystyle \frac{\mathrm{d}B(t)}{\mathrm{d}t}}\right|}^{\alpha }{\left|\mathrm{\Delta }B\right|}^{\beta -\alpha }\mathrm{d}t,$$

(33)

where $k_i$ is a coefficient determined by the Steinmetz parameters. In the general form,

$$k_i={\displaystyle \frac{k}{{(2\pi )}^{\alpha -1}{\displaystyle {\int }_0^{2\pi }|}\cos \theta {|}^{\alpha }{2}^{\beta -\alpha }\mathrm{d}\theta }}.$$

(34)

For a fair comparison, the classical SE, ModSE, iGSE, and the ISE model are calibrated on the same dataset. In each case, the unknown coefficients are identified by a differential evolution algorithm, and the resulting parameter values are summarized in

Table 4. Comparison of original and corrected equation parameters.

	k	α	β	τ	δ
Classical SE	2.397	1.366	2.128	-	-
ISE	4.119	1.402	2.154	−0.257	−3.085
ModSE	0.003	1.913	2.335	-	-
iGSE	0.002	1.953	2.344	-	-

. The optimization is carried out by minimizing a logarithmic-domain mean-squared error between the predicted and measured core losses:

$$F_{\log }(\theta )={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left[{\log }_{10}{P}_{\mathrm{pred},i}(\theta )-{\log }_{10}{P}_{\mathrm{meas},i}\right]}^2},$$

(35)

where $P_{\mathrm{pred},i}(\theta )$ and $P_{\mathrm{meas},i}$ denote the predicted and measured core losses, respectively. This definition is equivalent to performing a least-squares fit in the log-log space, which is common for Steinmetz-type models. The convergence behavior of the optimization process for the classical SE, ModSE, iGSE, and the ISE model is illustrated in Figure 8a–d, where the vertical axis shows the logarithmic scale of the dimensionless fitness $F_{\log }$. For the classical SE, the initial best fitness is of the order of 10¹, and it drops below 10⁻¹ within roughly the first 50 generations. After about 1500–1800 generations, the curve becomes almost flat, and the best fitness stabilizes at approximately 2.36 × 10⁻². For the ISE, the initial best fitness is below 10¹, but the larger parameter space leads to several pronounced decreases: between about the 1000th and 2000th generations, the best fitness is reduced from around 10⁰ to below 10⁻¹, and after approximately 2500 generations, it converges to a final value close to 0.93 × 10⁻². In other words, the improved model attains a best fitness that is about a factor of two smaller than that of the classical equation in the logarithmic domain, corresponding to a noticeably lower relative error in core-loss prediction. This confirms that the proposed temperature, harmonic, and high-frequency correction terms significantly enhance the prediction accuracy of the core loss model.

The convergence curve of the ModSE model is shown in Figure 8c. At the beginning of the evolution, the objective function decreases rapidly from a relatively high initial level. In the subsequent stage (around 200–400 generations), the best fitness exhibits a stepwise, slowly decreasing trend. After roughly 7 × 10² generations, the curve becomes nearly horizontal and converges to a value on the order of 2 × 10⁻². Overall, ModSE clearly reduces the fitting error compared with the classical SE. The iterative behavior of iGSE is presented in Figure 8d. Similarly to ModSE, a rapid decrease in the objective function is observed during the early iterations, but the descent is smoother and more persistent, with a final fitness lower than that of ModSE. The ISE model has the lowest final objective value among all four models. It should be noted that both ModSE and iGSE require additional high-resolution time-domain flux-density data (e.g., $B(t)$ waveforms) during fitting, which increases the amount of data to be acquired and the overall computational complexity compared with the ISE model.

To intuitively compare the prediction performance and distribution characteristics of the traditional and ISE, Gaussian kernel density estimation is used to plot the scatter of the actual magnetic core loss observed value and the traditional and ISE, respectively, as shown in Figure 9, which reflects the concentration of the sample points per unit area and illustrates the distribution density of the data. As shown in Figure 9, the color gradient reflects the distribution density of the scatter, blue indicates sparse and red indicates dense, and the dashed line indicates the ideal 1:1 correspondence, and the solid red line is the fitted regression line of the scatter. It can be seen that the prediction results obtained by the traditional SE show obvious dispersion, the dotted deviation between the scatter plot and the ideal state, and the fitted regression line also deviates from the ideal 1:1 contour line. In contrast, the scatter plot for the prediction results of the ISE (as shown in Figure 9b) is clearly concentrated near the dashed line, the fitted regression line in red has a similar trend, and R² improves by 5.18%. Thus, the ISE demonstrates good performance in predicting core loss and can provide reliable prior knowledge and feature support for the PINN-based model through physical constraints and explanatory forces.

For a quantitative evaluation of the prediction performance of the different models, four error metrics were adopted: normalized root mean squared error (NRMSE), normalized mean absolute error (NMAE), MAPE, and R². NRMSE and NMAE are used to provide scale-independent measures of error, so that models can be compared fairly across materials and operating conditions with different core-loss magnitudes. The results are summarized in

Table 5. Quantitative performance comparison of different Steinmetz-type loss models.

	Indicator	NRMSE	NMAE	MAPE	R2
Module
SE		0.0269	0.0139	0.3430	0.9353
ModSE		0.0258	0.0133	0.3335	0.9407
iGSE		0.0243	0.0125	0.3236	0.9489
ISE		0.0231	0.0120	0.3115	0.9532

. The classical SE model exhibits the weakest predictive performance, with an R² of 0.9353. The ModSE and iGSE models, which progressively incorporate waveform-related corrections, show improved accuracy, and their R² values increase to 0.9407 and 0.9489, respectively. In contrast, the proposed ISE model achieves the best performance across all four indicators, with NRMSE reduced to 0.0231, NMAE to 0.0120, MAPE to 0.3115, and R² increased to 0.9532. These results indicate that explicitly accounting for temperature effects, harmonic power ratios, and high-frequency correction terms enables a more accurate description of the physical characteristics of magnetic-core loss under complex excitation conditions.

4.4. Prediction Results of Magnetic Core Loss

Instead of splitting the data under a single operating condition, all samples were first labeled using the joint condition tuple. The dataset was then randomly shuffled while approximately preserving the overall condition distribution and subsequently divided into training, validation, and test subsets with a ratio of 75%/15%/10%. The same partition was additionally conducted within each material-and-condition subset. This procedure ensured that sinusoidal samples collected under identical temperature and frequency settings did not appear simultaneously in both the training and test sets, thereby reducing the risk of information leakage at the source.

For each sample, the original 1024-point time-domain flux-density sequence was converted into time-frequency descriptors through statistical feature extraction and PSD analysis. The resulting feature set comprised time-domain statistical descriptors and frequency-domain spectral features extracted from the flux-density waveform. Categorical variables, including material type and excitation waveform, were represented using one-hot encoding. Continuous variables were standardized using scaling parameters estimated exclusively from the training set, and the same transformation was then applied to the validation and test sets. Throughout this workflow, the test set was not used for parameter estimation or model selection.

As a complementary, data-driven representation, the Conv-LSTM module received the flux-density waveform reshaped into a four-dimensional tensor of shape (1, 1024, 1, 1), where each sample corresponds to a 1024-point time series with a single spatial dimension and a single channel. The first layer consists of a Conv-LSTM layer with 64 filters and a kernel size of (3, 1), enabling local temporal feature extraction while capturing long-range dependencies through LSTM gating. Padding is applied to preserve input–output dimensional consistency. The resulting feature maps are flattened and passed through two fully connected layers with 64 and 32 neurons, respectively, using ReLU activation to introduce nonlinearity, and a final output layer generates a single core-loss prediction. Model training is performed using the Adam optimizer with MSE as the optimization objective for 200 epochs and a batch size of 32. The main hyperparameters (including the weights of loss term in the loss function) are shown in

Table A1. Hyperparameters of the PINN-based model.

Module	Parameter	Value
Conv-LSTM block	Input sequence length	1024 samples
	Input shape	(1, 1024, 1, 1)
	Conv-LSTM filters	64
	Conv-LSTM kernel size	3 × 1
	Dense layers	64, 32
	Activation function	ReLU
	Optimizer	Adam
	Loss (data term)	MSE
	Batch size/Epochs	32/200
Loss function	MSE weight ${\lambda }_1$	0.2
	MAE weight ${\lambda }_2$	0.4
	Physics term weight ${\lambda }_3$	0.3
	Weight search range	[0, 1] with step 0.1
RF	Number of trees	300
	Max depth	10
	Min samples per split	2
	Min samples per leaf	2
	Max features	1.0
XGB	Number of trees	400
	Learning rate	0.1
	Max depth	6
	Subsample ratio	0.6
	Column subsample ratio	0.7
	$L_2$ regularization	1.0
GBR	Number of boosting stages	300
	Learning rate	0.1
	Max depth	6
	Subsample ratio	0.8
Stacking meta-learner	Model type	Linear regression
	Input features	${\widehat{y}}_{\mathrm{RF}},{\widehat{y}}_{\mathrm{XGB}},{\widehat{y}}_{\mathrm{GBR}}$
	Training data for coefficients	5-fold CV
	Linear regression coefficients	$w_0=-1487$, $w_1=0.423$, $w_2=0.453$, $w_3=0.124$

of Appendix A.

Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 illustrate the predictive performance of various modules of the PINN-based model, including the Conv-LSTM-PSD (Fusion) module, the GBR module after two rounds of feature fusion, the RF module, the XGB module, and the ensemble learning module integrating these modules. Among them, Figure 10a shows the comparison of the true magnetic core loss and the predicted values from the Fusion module reveal significant deviations, indicating the limitations of the module in accurately capturing the characteristics of magnetic core loss. Figure 10b shows the visualization of the regression analysis results where the black dashed line represents the ideal prediction scenario (i.e., perfect alignment of predicted and true values). However, the red solid line deviates substantially from the ideal line, indicating that the Fusion module has significant prediction errors. These results reveal that the Fusion module has limited predictive capability under complex conditions and requires further improvement. Moreover, Figure 11, Figure 12, and Figure 13 show the prediction results of the GBR, RF, and XGB modules, respectively. It can be seen from Figure 11a, Figure 12a, and Figure 13a that the RF and XGB modules demonstrate better alignment between predicted and true values compared to the GBR module, suggesting higher prediction accuracy. In the regression results of those modules, Figure 13b illustrates the regression results of the XGB module for magnetic core loss. It can be observed that the red regression line closely aligns with the ideal line (black dashed line), indicating minimal deviations between the predicted and true values. This phenomenon demonstrates that the XGB module exhibits high accuracy in the prediction of magnetic core loss. As shown in Figure 12b, the regression line of the RF module aligns less closely with the ideal line compared to the XGB module, although it still shows relatively small deviations. Furthermore, Figure 14 presents the prediction results of the complete PINN-based model, in which GBR, RF, and XGB modules are integrated through an ensemble learning framework after feature extraction and two rounds of feature fusion. As shown in Figure 14a, the prediction values are nearly identical to the true values, with minimal deviations observed across the entire sample range. As shown in Figure 14b, the scatter points are densely clustered around the ideal line, and the regression line almost perfectly overlaps with it. These results confirm that the ensemble learning module significantly outperforms individual modules in terms of predictive accuracy and robustness.

To further illustrate the prediction performance and error distribution of the GBR, RF, XGB, and Ensemble modules of the PINN-based model, Figure 15 compares the absolute errors of these modules, including the Fusion module, in the prediction of the magnetic core loss of the magnetic element. It can be observed that the Fusion module exhibits the largest error fluctuations, reflecting its limitations in achieving high prediction accuracy and robustness. In contrast, the GBR, RF, and XGB modules show relatively smaller errors and more stable fluctuations, indicating their improved prediction performance. Among all modules, the Ensemble module of the PINN-based model demonstrates the smallest absolute errors and the most stable fluctuations, highlighting its ability to effectively integrate the strengths of multiple modules. This significantly enhances prediction accuracy and robustness. These findings emphasize the superiority of the hybrid approach, which combines physics-driven and data-driven methodologies to deliver high accuracy and stability under complex conditions.

This section also uses the same error metrics (NRMSE, NMAE, MAPE, and R²) as mentioned earlier, with the focus on comparing the performance of different prediction modules across various operating conditions.

Table 6. Performance comparison of various prediction modules based on evaluation indicators.

	Indicator	NRMSE	NMAE	MAPE	R2
Module
PSD		18.0419	7.7501	7.4159	0.4639
Conv-LSTM		11.2137	4.7953	3.2958	0.6773
Fusion		7.6551	3.2750	2.8608	0.7834
GBR		4.3936	2.0476	2.0477	0.8854
RF		2.1693	1.0170	0.7917	0.9511
XGB		2.0803	0.9789	0.7178	0.9534
Ensemble (without Physical Priors)		2.2675	1.0437	0.7831	0.9496
Ensemble		1.9591	0.8017	0.6546	0.9603

summarizes the normalized performance indicators of the different prediction modules. And consistent with the trends in Figure 16, the purely spectral PSD-based model shows the largest errors, with NRMSE and NMAE of 18.04 and 7.75, and an R² of only 0.4639, indicating that the handcrafted spectral descriptors alone are insufficient to capture the nonlinear core-loss behavior. When the Conv-LSTM network is applied directly to the time-domain waveform, all four indicators are significantly improved (NRMSE = 11.21, NMAE = 4.80, MAPE = 3.30, R² = 0.6773), demonstrating the benefit of temporal feature learning. The Fusion module, which combines PSD features with Conv-LSTM representations, further reduces the errors (NRMSE = 7.66, NMAE = 3.28, MAPE = 2.86, R² = 0.7834), suggesting that the two feature types provide complementary information.

Among the three tree-based regressors, GBR already outperforms the purely spectral and Conv-LSTM modules (NRMSE = 4.39, NMAE = 2.05, MAPE = 2.05, R² = 0.8854), while RF and XGB deliver substantially better accuracy (RF: NRMSE = 2.17, NMAE = 1.02, MAPE = 0.79, R² = 0.9511; XGB: NRMSE = 2.08, NMAE = 0.98, MAPE = 0.72, R² = 0.9534). The full stacking Ensemble attains the best overall performance, with NRMSE, NMAE, and MAPE reduced to 1.96, 0.80, and 0.65, respectively, and R² increased to 0.9603. These results confirm that the ensemble learning framework effectively exploits the complementary strengths of the individual base learners.

To further clarify the role of the physics-informed prior, an ablation variant denoted “Ensemble (Without Physical Priors)” was constructed by removing the physics-based constraint term from the loss function and training the ensemble purely in a data-driven manner. As shown in Table 6, this variant still outperforms the individual RF, XGB, and GBR models, but its normalized errors are consistently higher than those of the full Ensemble (NRMSE = 2.27 vs. 1.96; NMAE = 1.04 vs. 0.80; MAPE = 0.78 vs. 0.65), and its R² is lower (0.9496 vs. 0.9603). This ablation study demonstrates that the performance gain of the proposed method does not come solely from the complex ensemble structure; the incorporation of the physical constraint provides an obvious improvement in accuracy and generalization.

In addition, to enhance the interpretability of the proposed ensemble, a feature-importance analysis was carried out based on the tree-based regressors. Figure 17 shows the ranked importance scores of the top 15 input variables. The most influential features are the time-domain descriptors skewness, energy, and peak value, followed by spectral energy, standard deviation, and peak-to-peak value. Together, these six variables already contribute 56.4% of the total importance, indicating that the overall loss behavior is strongly governed by the magnitude and shape of the flux-density waveform. When PSD_PCA_1, Kurtosis, Harmonic order, Conv_PCA_1, and Entropy spectral are included, the cumulative contribution of the top 10 features increases to 74.5%. The remaining features, Frequency, ExtraFeature_10, Channel bandwidth, and Temperature, still provide non-negligible information, but their individual importance scores are comparatively lower. Overall, the top 15 variables account for 84.8% of the total feature importance, suggesting that a compact subset of time-domain statistics and spectral descriptors is sufficient to capture most of the variance relevant to core-loss prediction. This ranking is also physically reasonable, since waveform asymmetry, energy content, and spectral distribution are known to have a dominant impact on magnetic-core loss under complex excitation conditions.

5. Conclusions

This study proposes a physics-informed neural network (PINN)-based model for accurate prediction of magnetic core loss of magnetic elements, which is designed to address the limitations of traditional methods under complex conditions, such as high temperature, non-sinusoidal waveform, and high frequency. The developed PINN-based model is constructed with a hybrid network architecture of a convolutional long short-term memory network (Conv-LSTM), power spectral density (PSD), and an ensemble learning method (extreme gradient boosting (XGB), gradient boosting regression (GBR), and random forest (RF)). This design aims to achieve effective feature fusion and nonlinear modeling. In particular, the feature extraction stage will process data from three dimensions: time domain, frequency domain, and material properties, with Conv-LSTM and PSD modules in the designed network architecture. The Steinmetz equation (SE) is improved by introducing temperature, waveform, and high-frequency correction terms to expand the applicability under complex conditions, and this improved Steinmetz equation (ISE) is integrated as physical constraints embedded in the neural network for the magnetic core loss prediction. By incorporating the physical laws into the loss function design, this model is required to follow the underlying physical laws during the data-driven learning process. The validity of the PINN-based model has been verified through a case study. Results showed that the prediction results closely match the trends of the actual values, with high consistency observed across various conditions. In terms of quantitative indicators, the PINN-based model achieves significant improvements, with the prediction error reduced by more than 30% compared to the classical SE. For example, the RMSE and MAE values of the model are 7.475 × 10⁴ and 3.156 × 10⁴, respectively, outperforming both single machine learning models and the classical SE. These results demonstrate that the proposed PINN-based model has high prediction accuracy and applicability under complex conditions.